The two leaders agreed to work towards a treaty to replace the 1991 Start-1 pact which expires in December and to reduce their stockpiles of nuclear warheads to between 1,500 and 1,675 each. In a further sign of its willingness to repair fractured relations, Russia offered its airspace to US planes flying weapons and troops to Afghanistan. This will shorten flying time and save the US more than $100m (£62m) a year.

Mr Obama announced that he intended to host a global nuclear summit next year to combat nuclear proliferation, the biggest threat to global security, and said he and Mr Medvedev had discussed a plan for Russia to reciprocate by hosting a follow-up meeting.

There were smiles all round as Mr Obama arrived for the summit which continues today. “We resolve to reset US-Russian relations so that we can co-operate more effectively in areas of common interest,” Mr Obama said at the Kremlin.

How much effect the bonhomie and toned-down rhetoric will have on still thorny issues such as the proposed US missile defence system and the situation in Georgia remains to be seen. Even the nuclear agreement is only a framework for negotiators to use when drawing up a comprehensive treaty. Differences in opinion remain.

But Mr Obama and Mr Medvedev were full of positive words about each other and said they looked forward to continuing their working relationship.

Russia insisted before the talks took place that any negotiations on arms reduction had to proceed together with negotiations on the US plans for an anti-missile shield in central and eastern Europe, a Bush-era plan to which the Kremlin is strongly opposed.

There were no breakthroughs on this issue, though Mr Obama appeared to cede some ground. He said he thought it was inappropriate to link missile defence with arms reduction but Mr Medvedev announced they had “agreed that defensive and offensive weapons should be viewed as a whole”.

Mr Obama said: “We have agreed that we’re going to continue to discuss it. It’s going to take some hard work because it requires breaking down some long-standing suspicions”. He promised to present a review of the US plans by the end of the summer. Russia will be pleased at the change of tone ? the Bush administration had simply stated that the missile defence plans were none of Russia’s business.

Air Force One touched down in Moscow at lunchtime. Mr Obama, his wife, Michelle, and their daughters were ferried to the city centre, where Mr Obama laid a wreath at the Tomb of the Unknown Soldier. He then proceeded to the Kremlin to meet Mr Medvedev. As Mr Obama talked with his Russian counterpart for nearly four hours, his wife and daughters were given a tour of the Kremlin by Svetlana Medvedeva, the Russian President’s wife.

Mr Obama’s visit has not been given much airtime by the government-controlled media and his arrival in Moscow did not provoke the sort of public enthusiasm that has greeted him on other foreign visits.

The real test for renewed relations could come today, when Mr Obama has breakfast with Vladimir Putin, Russia’s former president, who as Prime Minister is still seen as the country’s most powerful man.

Mr Putin may have some of his acerbic wit at the ready for the President after Mr Obama made the curious step of publicly criticising Mr Putin before his trip to Moscow.

While praising Mr Medvedev as someone with whom it was possible to do business, Mr Obama said late last week that Mr Putin needed to understand that “the old Cold War approach to US-Russian relations is outdated” and that unlike Mr Medvedev, the former KGB agent who was now Prime Minister “has one foot in the old ways of doing business”. Mr Putin’s response was surprisingly measured but he is not known to take criticism well so Mr Obama’s attack may yet backfire.

Yesterday, Mr Obama looked awkward when answering a question at a press conference on who he felt held real power in Russia. He said his main interest was in “dealing directly with my counterpart” who was “straightforward and professional” but he also wanted to “reach out to Prime Minister Putin and other influential sectors of society”. Mr Medvedev gave a faint smile. Later, Mr Obama inadvertently began to refer to “President Putin”, but quickly corrected himself.

Mr Obama has gone out of his way to praise Mr Medvedev. In an interview published yesterday with the opposition newspaper Novaya Gazeta, he dodged a question about whether he would press Russian authorities over bringing to justice the murderers of the investigative journalist Anna Politkovskaya.

He also said it seemed “odd” that the former Yukos boss Mikhail Khodorkovsky was being subjected to a second trial in Russia but added that there was no place for outsiders to interfere in the Russian legal system and praised Mr Medvedev’s “courageous initiative to strengthen the rule of law in Russia”.

Most analysts feel that Mr Medvedev’s words about legal reform have not been followed up with substantive action.

In addition to his breakfast with Mr Putin today, Mr Obama is also due to give a substantial speech on US-Russia relations at the graduation ceremony of the New Economic School in Moscow.

He will also meet opposition leaders, including controversial radical opposition figures such as the former chess great Garry Kasparov. He departs Russia for the G8 summit in L’Aquila tomorrow morning.